Two fundamentally different approaches to education compete for acceptance. The first features the direct transfer of knowledge from master to student—using lecture, reading or demonstration. In the second, the student is guided in an exploratory process during which knowledge is personally discovered. The second method, advocated by Socrates, Maria Montessori and most graduate school programs, claims to achieve deeper understanding and higher retention. But its success depends on the creativity and energy of teachers as they guide individual students. In order to reduce this demand, many tools have been developed to promote the students' discovery of knowledge.
Mathematics education has a history of exploratory learning. A progression of manipulative devices have transformed mathematical abstractions into arrangements of concrete objects made of wood, bone, metal or polymer. Some of these illustrate the properties of numbers, others allow the student to explore operations. Some manipulatives (such as the abacus and the slide rule) demonstrate utility beyond learning and enter the everyday toolbox of those who work with numbers.
Among the aspirations of the present invention is to combine the proven educational value of a math manipulative with the logic engine of a computer algebra systems. These may form a virtual manipulative in which displayed symbols emulate the physicality of material tokens while obeying formal rules.
The roots of the present invention predate the digital computer and the modern patent system. Manipulative devices have been developed to aid math learning and performance since civilization made calculation necessary.
Four millenia ago, the abacus appears. Variations in design support calculation in cultures that use base 60 (Sumerian), base 20 (Aztec) as well as the ubiquitous base 10 number systems. The abacus is a powerful tool for arithmetic processing, but it does little to support symbolic thinking.
An alternative to the abacus was the ancient Chinese system of counting rods. In the 16th century, when the abacus finally displaced them, counting rods were adapted by Japanese algebrist Seki Takakazi as physical tokens that served to introduce his system of symbolic manipulation within the isolated mathematic tradition of wasan.
The slide rule, invented in 18th century Europe, is a familiar manipulative. Unlike the abacus, with its discrete states, the slide rule is an analog computer. It was used for calculation until replaced by the electronic digital calculator in the late 20th century. Although it had a place in schools, the slide rule was principally used for practical engineering. The slide rule offers its user power over numbers, even irrational numbers—but not abstract symbols.
In the twentieth century, classroom manipulatives were developed specifically to expose young children to basic arithmetic and the properties of the base 10 numbers. Most fundamental of these are Cuisenaire rods (1952), colored rods with integer lengths which introduce arithmetic, fractions, and squares.
Sellon (U.S. Pat. No. 4,445,865, 1984) improves the Cuisenaire system to better teach base 10 multiplication. Patents for similar grid-like games, based on familiar multiplication tables, stretch back to Verneau (U.S. Pat. No. 381,829.1922).
Rainey (1998) teaches a device for manipulating base 10 numbers. Such manipulatives teach concrete number relationships—but do not help students learn symbolic thinking.
Not all manipulatives involve rods or grids: Lewis (1926) teaches a reconfigurable roulette apparatus for educational games. While it can be applied to mathematics, it assists only in rote memorization and simple drills.
More relevantly, Donecker (1905) produces a tool for learning algebra. Using a system of balance beams and pans, it builds a physical analog for certain linear equations. The device illustrates the laws of proportionality, and provides a concrete sense of the unknown variable. However it offers no insight into symbolic reasoning.
Harder (1980) presents a four-player game of competitive algebraic solution. His board game allows learners to share the experience of solving an equation. Equations (and inequalities) are presented on wooden strips. Using familiar wooden tiles, the players take turns solving the equations. The board rotates 90 degrees at each turn, with each player validating the work of the previous player. Points are scored for correct operations and solving equations.
The Harder game provides limited exercise in symbolic manipulation, but does so without the benefit of technology.
The introduction of the electronic digital computer allows the evolution of logical systems not limited to manipulation of numbers. Symbolic logic is an early concern of computer scientists. An application providing this capability is called a Computer Algebra System (CAS).
The seminal CAS programs predate the patentability of software. Among these are Macsyma, begun in 1968 at MIT, and released commercially in 1982. At CalTech, SMP and Schoonschip (1963) were the antecedents of the commercially successful program Mathematica (1988, Wolfram) which shares market domination with Maple (1980, University of Waterloo).
Like most CAS packages, Maple and Mathematica are solvers—not teaching programs. They do not demonstrate the internal steps required to go from problem to solution.
Handheld calculators have been marketed which include Computer Algebra Systems. Prominent among these is the TI-NSpire CAS (2006) which has an embedded version of Derive (1988), a small-footprint CAS written, like many algebraic solvers, in a form of LISP.
The prior art includes several examples of interfaces built on top of Computer Algebra Systems with the intent of providing a learning experience.
Certain packages, such as the Algebrator (SoftMath, 1990?) explicitly show every step of the transition, and provide textual explanation of each of these steps. This is meant to educate the student, but it offers no exploratory pathways. Marketed to students as an automated tutor, it could also be abused as a homework robot.
Bonadio (1993) teaches a Computer Algebra System in which a graphic user interface (GUI) allows users to enter commands. Bonadio's system, known variously as Theorist, MathView and LiveMath employs a GUI to direct the CAS to derive a new expression from the existing expression, and to display the new one below the old one. A sequence of expressions cascade down the screen. By displaying line after line of expressions, in the manner of a console-driven application, the software does not behave like a manipulative device. Further, Bonadio's device offers no algebraic response to the user's actions until the user had completely specified the operation.
Addressing a different market, Razdow (1994) introduces the concept of a “live” symbolic expression. Using his software, MathSoft 3.1, a user can manipulate an expression in a document and rely on the editor's symbolic algebra engine to alter other expressions so that they remain true. This invention can be seen as combining the ‘live’ nature of a spreadsheet with the graphic sophistication of an equation compositor. The goal is exposition, not education. The actual solutions that MathSoft performs go little beyond altering the initial assignments in a system of equations and performing the consequent substitutions.
Vernon (application 2004) demonstrates concern for the learning process, and introduces an interactive experience that monitors student performance. Given an expression, the student types in a line of text which represents the new expression after applying an algebraic operation, as a single step toward the solution. The software analyzes both initial and subsequent statements, and searches a rulebook to determine the validity of the student's work. Errors, of course, are flagged. The steps are repeated until the solution is arrived at.
Vernon's technological claims, which—like most CAS engines—focus on the nodal structure of expressions, are interesting in that they amount to a sophisticated equivalent of the “diff” program: they compare two expressions and remove the equivalences. The remaining elements are subjected to a search through a rulebook database.
Vernon's invention compels the student to iteratively enter ASCII text strings with incremental changes to the statement—rather than directly moving the terms of a classical math representation.
It only tests the validity of work after it is performed, rather than supporting the exploratory learner with hints, previews and suggestions.